There's a sibling comment saying it's due to the exposure, that's wrong. That's not how Hubble works, it's not just a giant digital camera in orbit. The "photo" you're looking at is a composite from tens or hundreds of other images, so to start with most of the pixels you're looking at are just black filler.
You can see the black and white raw images at http://hst.esac.esa.int/ehst/#home search for "SATURN" under "Solar System Objects" and sort by date. A lot of the Hubble images from June 20th, 2019 have stars, including some visible between the rings and Saturn. They were just edited out because showing them wasn't the point.
[0] http://hst.esac.esa.int/ehst-sl-server/servlet/data-action?R...
I'm referring to objects you can consistently see between different observations. E.g. this one around 1/2 a planet width from the pole you can't see observed from 2018-06-06 14:35:05 until 2018-06-06 14:46:35:
http://hst.esac.esa.int/ehst-sl-server/servlet/data-action?R...
http://hst.esac.esa.int/ehst-sl-server/servlet/data-action?R...
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Although looking again perhaps that's a moon of Saturn's rather than a particularly bright star.
Not an astronomer, just an interested amateur. The few times I've been curious enough to look at the raw data behind these images I've found background celestial objects that are definitely edited out. That's understandable, the point is to show e.g. Saturn and not whatever background objects happened to be there at the time Hubble took the photo.
I'm just pointing out it's not all due to the light exposure.
https://www.esa.int/spaceinimages/Images/2007/07/The_constel...
EDIT: A larger version of the image, with an equally huge url: https://m.esa.int/var/esa/storage/images/esa_multimedia/imag...
Even in a backyard telescope, if you look at Saturn, its brightness will wash out all but the very brightest of nearby stars because your eye adjusts for Saturn's brightness, not star brightness. (One nearby "star" you may see is actually the moon Titan.)
The New Horizons probe did pick up star images in Ultima-Thule images, if you look at the raw exposures, but Ultima-Thule is really far away from the sun, beyond Pluto. And it's a dark body, requiring sensitive camera settings. Note that some raw exposures from some probes will show streaks that may be mistaken for stars, but are actually cosmic rays. They are usually digitally removed from final versions.
The boundary where planetary exposures will start to clearly show stars is roughly around Neptune's or Pluto's orbit. But processing may have to remove them because time-lapse and/or multiple exposures may be needed to properly capture dim planets and rocky bodies that far out. Any stars caught would otherwise look smeared or smugdy.